Black-Scholes equation as one of the most celebrated mathematical models has an explicit analytical solution known as the Black-Scholes formula. Later variations of the equation, such as fractional or nonlinear Black-Scholes equations, do not have a closed form expression for the corresponding formula. In that case, one will need asymptotic expansions, including homotopy perturbation method, to give an approximate analytical solution. However, the solution is non-smooth at a special point. We modify the method by {first} performing variable transformations that push the point to infinity. As a test bed, we apply the method to the solvable Black-Scholes equation, where excellent agreement with the exact solution is obtained. We also extend our study to multi-asset basket and quanto options by reducing the cases to single-asset ones. Additionally we provide a novel analytical solution of the single-asset quanto option that is simple and different from the existing expression.

翻译：最有名的数学模型之一的黑球方程式有明确的分析解决方案,称为黑球方程式。后来的方程式变异,如分数或非线性黑球方程式等,对相应的公式没有封闭的表达形式。在这种情况下,人们需要无线扩张,包括同质扰动法,以提供大致的分析解决方案。然而,解决方案在一个特殊点是非吸附的。我们用{第一}修改方法,将点推向无限化。作为测试床,我们将方法应用到可溶解的黑球方程式,在其中取得与确切解决方案的极佳一致。我们还将研究扩展到多资产篮子和二次方程式选项,将案例简化为单资产方程式。此外,我们提供了简单和与现有表达方式不同的单资产方程式新分析解决方案。